QED Logic quod erat demonstrandum

An Expert 10×10 Puzzle Solved with Forced Chain in 37 Steps

10×10 · expert · 37 steps

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This 10×10 board rates as expert (difficulty score 1698). Solving it from scratch takes 37 logical steps, using 8 techniques: Single cell, Row/column exclusion, Adjacency clear, Region confinement, Line confinement, Paired regions, Shape squeeze, Forced chain. Every step below is forced: nothing here is a guess.

  1. First, every remaining candidate in this region touches B5, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  2. Next, every remaining candidate in this region touches B8, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  3. Then, marking A1 would force B7 (the last open cell in region 8), and then region 6 would be left with no open cell for its marker. So A1 can't be the marker there.
  4. After that, if B1 were the marker, it would force A5, A6 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C6 (the last open cell in region 6), then D9 (the last open cell in region 9), and the chain continues, and region 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
  5. Now, that one can't be the marker: at C1 it would force A10, B9 and B10 cleared (together, two regions have open cells only in columns A and B), then D9 (the last open cell in region 9), then F10 (the last open cell in region 10), and the chain continues, and region 5 would be left with no open cell for its marker.
  6. From there, that one can't be the marker: at A2 it would force B7 (the last open cell in region 8), and region 6 would be left with no open cell for its marker.
  7. Following that, marking B2 would force A5, A6 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C6 (the last open cell in region 6), then D9 (the last open cell in region 9), and the chain continues, and then region 7 would be left with no open cell for its marker. So B2 can't be the marker there.
  8. First, if C2 were the marker, it would force A10, B9 and B10 cleared (together, two regions have open cells only in columns A and B), then D9 (the last open cell in region 9), then F10 (the last open cell in region 10), and the chain continues, and column B would end up with no open cell for its marker, which can't happen. So it can't go there.
  9. Next, if A3 were the marker, it would force B7 (the last open cell in region 8), and region 6 would end up with no open cell for its marker, which can't happen. So it can't go there.
  10. Then, that one can't be the marker: at B3 it would force A5, A6 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C6 (the last open cell in region 6), then D9 (the last open cell in region 9), and the chain continues, and region 7 would be left with no open cell for its marker.
  11. After that, that one can't be the marker: at D2 it would force J1 (the last open cell in row 1), then G4, H4 and I4 cleared (every open cell left in this region sits in row 4, so its marker has to land there), then G5, H5, I5, and 5 more cleared (every open cell in row 3 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and column I would be left with no open cell for its marker.
  12. Now, if I2 were the marker, it would force D3, E3, E4, and 2 more cleared (every open cell in row 1 belongs to the same region, so that region's marker has to be somewhere in this row), then C3 (the last open cell in row 3), then G5, H5, J5, and 5 more cleared (every open cell in row 4 belongs to the same region, so that region's marker has to be somewhere in this row), and the chain continues, and row 8 would end up with no open cell for its marker, which can't happen. So it can't go there.
  13. From there, if D3 were the marker, it would force J1 (the last open cell in row 1), and row 2 would end up with no open cell for its marker, which can't happen. So it can't go there.
  14. Following that, that one can't be the marker: at E3 it would force J1 (the last open cell in row 1), and row 2 would be left with no open cell for its marker.
  15. First, marking F3 would immediately leave row 2 with no open cell for its marker, so it can't be the marker there.
  16. Next, if G3 were the marker, it would force E2 (the last open cell in row 2), and row 1 would end up with no open cell for its marker, which can't happen. So it can't go there.
  17. Then, marking A4 would force B7 (the last open cell in region 8), and then region 6 would be left with no open cell for its marker. So A4 can't be the marker there.
  18. After that, if B4 were the marker, it would force D6, E6, F6, and 4 more cleared (every open cell left in this region sits in row 6, so its marker has to land there), then A6 and A10 cleared (every open cell left in this region sits in column A, so its marker has to land there), then C6 (the last open cell in region 6), and the chain continues, and region 7 would end up with no open cell for its marker, which can't happen. So it can't go there.
  19. Now, every open cell left in this region sits in column C, so its marker has to land there; that clears every other open cell in the column.
  20. From there, together, two regions have open cells only in columns A and B; with one marker each, those two markers have to fill exactly those two columns, so every other region's open cells there can be cleared.
  21. Following that, place a marker at D9. Region has exactly one open cell left. Placing here clears the rest of row 9, column D, and its region. Placing here also clears its 2 touching neighbours.
  22. First, place a marker at F10. Region has exactly one open cell left. Placing here clears the rest of row 10, column F, and its region.
  23. Next, every open cell left in this region sits in column E, so its marker has to land there; that clears every other open cell in the column.
  24. Then, every open cell left in this region sits in row 1, so its marker has to land there; that clears every other open cell in the row.
  25. After that, every remaining candidate in this region touches H2, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  26. Now, every remaining candidate in this region touches H7, so whichever candidate ends up marked will clear it, and it can be ruled out now regardless of which one wins.
  27. From there, marking G1 would force J2 (the last open cell in region 3), then A6, B6 and E6 cleared (every open cell left in this region sits in row 6, so its marker has to land there), then A5 (the last open cell in region 6), and the chain continues, and then column B would be left with no open cell for its marker. So G1 can't be the marker there.
  28. Following that, that one can't be the marker: at I1 it would force G2 (the last open cell in region 3), then H6 (the last open cell in region 7), then A5 (the last open cell in region 6), and the chain continues, and column B would be left with no open cell for its marker.
  29. First, place a marker at H1. Region has exactly one open cell left. Placing here clears the rest of row 1, column H, and its region. Placing here also clears its 1 touching neighbour.
  30. Next, place a marker at J2. Region has exactly one open cell left. Placing here clears the rest of row 2, column J, and its region. Placing here also clears its 1 touching neighbour.
  31. Then, place a marker at C3. Row 3 has exactly one open cell left. Placing here clears the rest of row 3, column C, and its region.
  32. After that, every open cell in row 4 belongs to the same region, so that region's marker has to be somewhere in this row; that clears every other open cell in the region.
  33. Now, place a marker at A5. Row 5 has exactly one open cell left. Placing here clears the rest of row 5, column A, and its region.
  34. From there, place a marker at B7. Region has exactly one open cell left. Placing here clears the rest of row 7, column B, and its region.
  35. Following that, place a marker at E6. Region has exactly one open cell left. Placing here clears the rest of row 6, column E, and its region.
  36. First, place a marker at G8. Region has exactly one open cell left. Placing here clears the rest of row 8, column G, and its region.
  37. Finally, place a marker at I4. Region has exactly one open cell left.

The key move here was Forced chain. Once you can spot that, this board falls quickly. ∎